3-D electrical resistivity tomography using adaptive wavelet parameter grids

We present a novel adaptive model parametrization strategy for the 3-D electrical resistivity tomography problem and demonstrate its capabilities with a series of numerical examples. In contrast to traditional parametrization schemes, which are based on fixed disjoint blocks, we discretize the subsurface in terms of Haar wavelets and adaptively adjust the parametrization as the iterative inversion proceeds. This results in a favourable balance of cell sizes and parameter reliability, that is, in regions where the data constrain the subsurface properties well, our parametrization strategy leads to a fine grid, whereas poorly resolved areas are represented only by a few large blocks. This is documented with eigenvalue analyses and by computing model resolution matrices. During the initial iteration steps, only a few model parameters are involved, which reduces the risk that the regularization dominates the inversion. The algorithm also automatically accounts for non-linear effects caused by pronounced conductivity contrasts. Inside conductive features a finer grid is generated than inside more resistive structures. The automated parameter adaptation is computationally efficient, because the coarsening and refinement subroutines have a nearly linear numerical complexity with respect to the number of model parameters. Because our approach is not tightly coupled to electrical resistivity tomography, it should be straightforward to adapt it to other data type